cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.8%

Time: 18.2s

Alternatives: 7

Speedup: 34.9×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan \left(x \cdot 0.5\right) \cdot \sin x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0002)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (* (tan (* x 0.5)) (sin x)) x) x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0002) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((tan((x * 0.5)) * sin(x)) / x) / x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0002d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((tan((x * 0.5d0)) * sin(x)) / x) / x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0002) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((Math.tan((x * 0.5)) * Math.sin(x)) / x) / x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0002:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((math.tan((x * 0.5)) * math.sin(x)) / x) / x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0002)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(tan(Float64(x * 0.5)) * sin(x)) / x) / x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0002)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((tan((x * 0.5)) * sin(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0002], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 32.8%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 69.2%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 69.2%

         \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664 \]

   4. Simplified 69.2%

      \[\leadsto \color{blue}{0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664} \]

   if 2.0000000000000001e-4 < x

   1. Initial program 96.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. flip-- 96.4%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

      2. div-inv 96.4%

         \[\leadsto \frac{\color{blue}{\left(1 \cdot 1 - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]

      3. metadata-eval 96.4%

         \[\leadsto \frac{\left(\color{blue}{1} - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

      4. 1-sub-cos 96.4%

         \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

      5. pow2 96.4%

         \[\leadsto \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

   3. Applied egg-rr 96.4%

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]
   4. Step-by-step derivation

      1. unpow2 96.4%

         \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

      2. associate-*l* 96.5%

         \[\leadsto \frac{\color{blue}{\sin x \cdot \left(\sin x \cdot \frac{1}{1 + \cos x}\right)}}{x \cdot x} \]

      3. associate-*r/ 96.5%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]

      4. *-rgt-identity 96.5%

         \[\leadsto \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]

      5. hang-0p-tan 96.9%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x \cdot x} \]

   5. Simplified 96.9%

      \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x \cdot x} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 96.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\right)\right)} \]

      2. expm1-udef 58.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\right)} - 1} \]

      3. div-inv 58.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right) \cdot \frac{1}{x \cdot x}}\right)} - 1 \]

      4. div-inv 58.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}\right) \cdot \frac{1}{x \cdot x}\right)} - 1 \]

      5. metadata-eval 58.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(x \cdot \color{blue}{0.5}\right)\right) \cdot \frac{1}{x \cdot x}\right)} - 1 \]

      6. pow2 58.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)} - 1 \]

      7. pow-flip 58.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \color{blue}{{x}^{\left(-2\right)}}\right)} - 1 \]

      8. metadata-eval 58.8%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot {x}^{\color{blue}{-2}}\right)} - 1 \]

   7. Applied egg-rr 58.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot {x}^{-2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot {x}^{-2}\right)\right)} \]

      2. expm1-log1p 99.7%

         \[\leadsto \color{blue}{\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot {x}^{-2}} \]

      3. metadata-eval 99.7%

         \[\leadsto \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot {x}^{\color{blue}{\left(2 \cdot -1\right)}} \]

      4. pow-sqr 99.6%

         \[\leadsto \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \]

      5. unpow-1 99.6%

         \[\leadsto \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \]

      6. unpow-1 99.6%

         \[\leadsto \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \]

      7. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \frac{1}{x}\right) \cdot \frac{1}{x}} \]

      8. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot 1}{x}} \cdot \frac{1}{x} \]

      9. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\left(\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x} \cdot 1\right)} \cdot \frac{1}{x} \]

      10. *-rgt-identity 99.7%

          \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}} \cdot \frac{1}{x} \]

      11. associate-*r/ 99.7%

          \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x} \cdot 1}{x}} \]

      12. *-rgt-identity 99.7%

          \[\leadsto \frac{\color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}}}{x} \]

      13. associate-*r/ 99.7%

          \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}}}{x} \]

      14. associate-*r/ 99.7%

          \[\leadsto \color{blue}{\sin x \cdot \frac{\frac{\tan \left(x \cdot 0.5\right)}{x}}{x}} \]

      15. associate-/r* 96.8%

          \[\leadsto \sin x \cdot \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x \cdot x}} \]

      16. *-commutative 96.8%

          \[\leadsto \sin x \cdot \frac{\tan \color{blue}{\left(0.5 \cdot x\right)}}{x \cdot x} \]

   9. Simplified 96.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\tan \left(0.5 \cdot x\right)}{x \cdot x}} \]

   11. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\frac{\frac{\tan \left(0.5 \cdot x\right) \cdot \sin x}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan \left(x \cdot 0.5\right) \cdot \sin x}{x}}{x}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0046)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ 1.0 x) (/ x (- 1.0 (cos x))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0046) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0046d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 / x) / (x / (1.0d0 - cos(x)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0046) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x) / (x / (1.0 - Math.cos(x)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0046:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 / x) / (x / (1.0 - math.cos(x)))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0046)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 / x) / Float64(x / Float64(1.0 - cos(x))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0046)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0046], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(x / N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0045999999999999999

   1. Initial program 32.8%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 69.2%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 69.2%

         \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664 \]

   4. Simplified 69.2%

      \[\leadsto \color{blue}{0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664} \]

   if 0.0045999999999999999 < x

   1. Initial program 96.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. associate-/r* 99.3%

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

      2. div-inv 99.4%

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
   4. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}} \]

      2. clear-num 99.3%

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{1}{\frac{x}{1 - \cos x}}} \]

      3. un-div-inv 99.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0046)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0046) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0046d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0046) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0046:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0046)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0046)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0046], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0045999999999999999

   1. Initial program 32.8%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 69.2%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 69.2%

         \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664 \]

   4. Simplified 69.2%

      \[\leadsto \color{blue}{0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664} \]

   if 0.0045999999999999999 < x

   1. Initial program 96.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0046)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0046) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0046d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0046) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0046:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0046)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0046)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0046], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0045999999999999999

   1. Initial program 32.8%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 69.2%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 69.2%

         \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664 \]

   4. Simplified 69.2%

      \[\leadsto \color{blue}{0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664} \]

   if 0.0045999999999999999 < x

   1. Initial program 96.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. associate-/r* 99.3%

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

      2. div-inv 99.4%

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
   4. Step-by-step derivation

      1. un-div-inv 99.3%

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 5: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.4)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* 2.0 (pow x -2.0))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 2.0 * pow(x, -2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = 2.0d0 * (x ** (-2.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 2.0 * Math.pow(x, -2.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = 2.0 * math.pow(x, -2.0)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(2.0 * (x ^ -2.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = 2.0 * (x ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.4], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 32.8%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 69.2%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 69.2%

         \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664 \]

   4. Simplified 69.2%

      \[\leadsto \color{blue}{0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664} \]

   if 2.39999999999999991 < x

   1. Initial program 96.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. div-sub 96.4%

         \[\leadsto \color{blue}{\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}} \]

      2. pow2 96.4%

         \[\leadsto \frac{1}{\color{blue}{{x}^{2}}} - \frac{\cos x}{x \cdot x} \]

      3. pow-flip 96.2%

         \[\leadsto \color{blue}{{x}^{\left(-2\right)}} - \frac{\cos x}{x \cdot x} \]

      4. metadata-eval 96.2%

         \[\leadsto {x}^{\color{blue}{-2}} - \frac{\cos x}{x \cdot x} \]

      5. div-inv 96.4%

         \[\leadsto {x}^{-2} - \color{blue}{\cos x \cdot \frac{1}{x \cdot x}} \]

      6. pow2 96.4%

         \[\leadsto {x}^{-2} - \cos x \cdot \frac{1}{\color{blue}{{x}^{2}}} \]

      7. pow-flip 99.3%

         \[\leadsto {x}^{-2} - \cos x \cdot \color{blue}{{x}^{\left(-2\right)}} \]

      8. metadata-eval 99.3%

         \[\leadsto {x}^{-2} - \cos x \cdot {x}^{\color{blue}{-2}} \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{{x}^{-2} - \cos x \cdot {x}^{-2}} \]

   4. Taylor expanded in x around 0 58.8%

      \[\leadsto {x}^{-2} - \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 58.8%

         \[\leadsto {x}^{-2} - \frac{1}{\color{blue}{x \cdot x}} \]

   6. Simplified 58.8%

      \[\leadsto {x}^{-2} - \color{blue}{\frac{1}{x \cdot x}} \]

   8. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{{x}^{-2} + {x}^{-2}} \]
   9. Step-by-step derivation

      1. count-2 63.8%

         \[\leadsto \color{blue}{2 \cdot {x}^{-2}} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{2 \cdot {x}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-2}\\ \end{array} \]

Alternative 6: 75.6% accurate, 34.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.5e+77) 0.5 0.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.5e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.5e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.5e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.5e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.5e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e77

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 65.8%

      \[\leadsto \color{blue}{0.5} \]

   if 1.4999999999999999e77 < x

   1. Initial program 96.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. div-sub 95.9%

         \[\leadsto \color{blue}{\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}} \]

      2. pow2 95.9%

         \[\leadsto \frac{1}{\color{blue}{{x}^{2}}} - \frac{\cos x}{x \cdot x} \]

      3. pow-flip 95.7%

         \[\leadsto \color{blue}{{x}^{\left(-2\right)}} - \frac{\cos x}{x \cdot x} \]

      4. metadata-eval 95.7%

         \[\leadsto {x}^{\color{blue}{-2}} - \frac{\cos x}{x \cdot x} \]

      5. div-inv 95.8%

         \[\leadsto {x}^{-2} - \color{blue}{\cos x \cdot \frac{1}{x \cdot x}} \]

      6. pow2 95.8%

         \[\leadsto {x}^{-2} - \cos x \cdot \frac{1}{\color{blue}{{x}^{2}}} \]

      7. pow-flip 99.5%

         \[\leadsto {x}^{-2} - \cos x \cdot \color{blue}{{x}^{\left(-2\right)}} \]

      8. metadata-eval 99.5%

         \[\leadsto {x}^{-2} - \cos x \cdot {x}^{\color{blue}{-2}} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{{x}^{-2} - \cos x \cdot {x}^{-2}} \]

   4. Taylor expanded in x around 0 73.6%

      \[\leadsto {x}^{-2} - \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 73.6%

         \[\leadsto {x}^{-2} - \frac{1}{\color{blue}{x \cdot x}} \]

   6. Simplified 73.6%

      \[\leadsto {x}^{-2} - \color{blue}{\frac{1}{x \cdot x}} \]

   7. Taylor expanded in x around 0 73.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 27.5% accurate, 107.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 0.0)

C

x = abs(x);
double code(double x) {
	return 0.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 0.0;
}

Python

x = abs(x)
def code(x):
	return 0.0

Julia

x = abs(x)
function code(x)
	return 0.0
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 0.0;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 0.0

Derivation

1. Initial program 48.0%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Step-by-step derivation

   1. div-sub 48.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}} \]

   2. pow2 48.5%

      \[\leadsto \frac{1}{\color{blue}{{x}^{2}}} - \frac{\cos x}{x \cdot x} \]

   3. pow-flip 48.5%

      \[\leadsto \color{blue}{{x}^{\left(-2\right)}} - \frac{\cos x}{x \cdot x} \]

   4. metadata-eval 48.5%

      \[\leadsto {x}^{\color{blue}{-2}} - \frac{\cos x}{x \cdot x} \]

   5. div-inv 48.5%

      \[\leadsto {x}^{-2} - \color{blue}{\cos x \cdot \frac{1}{x \cdot x}} \]

   6. pow2 48.5%

      \[\leadsto {x}^{-2} - \cos x \cdot \frac{1}{\color{blue}{{x}^{2}}} \]

   7. pow-flip 49.3%

      \[\leadsto {x}^{-2} - \cos x \cdot \color{blue}{{x}^{\left(-2\right)}} \]

   8. metadata-eval 49.3%

      \[\leadsto {x}^{-2} - \cos x \cdot {x}^{\color{blue}{-2}} \]

3. Applied egg-rr 49.3%

   \[\leadsto \color{blue}{{x}^{-2} - \cos x \cdot {x}^{-2}} \]

4. Taylor expanded in x around 0 26.4%

   \[\leadsto {x}^{-2} - \color{blue}{\frac{1}{{x}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 26.4%

      \[\leadsto {x}^{-2} - \frac{1}{\color{blue}{x \cdot x}} \]

6. Simplified 26.4%

   \[\leadsto {x}^{-2} - \color{blue}{\frac{1}{x \cdot x}} \]

7. Taylor expanded in x around 0 27.2%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 27.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))